PreCalculus Alignment Record
|Know and use a definition of a function to decide if a given relation is a function
|Perform algebraic operations (including
compositions) on functions and apply transformations
(translations, reflections, and rescalings).
|Write an expression for the composition of one
given function with another and find the domain,
range, and graph of the composite function. Recognize components when a function is composed of
two or more elementary functions
|Determine whether a function (given symbolically
or graphically) has an inverse and express the
inverse (symbolically, if the function is given symbolically, or graphically, if given graphically) if it
exists. Know and interpret the function notation for inverses.
|Determine whether two given functions are inverses, using composition.
|Identify and describe discontinuities of a
function (e.g., greatest integer function, 1/x) and how these
relate to the graph.
|Understand the concept of limit of a function as
x approaches a number or infinity. Use the idea of
limit to analyze a graph as it approaches an asymptote. Compute limits of simple functions (e.g., find
the limit as x approaches 0 of f(x) = 1/x) informally.
|Explain how the rates of change of functions in
different families (e.g., linear functions, exponential
functions, etc.) differ, referring to graphical representations.
|Exponential and Logarithmic Functions
|Use the inverse relationship between exponential
and logarithmic functions to solve equations and
|Graph logarithmic functions. Graph translations and reflections of these functions.
|Compare the large-scale behavior of exponential
and logarithmic functions with different bases and
recognize that different growth rates are visible in the graphs of the functions.
|Solve exponential and logarithmic equations when
possible , (e.g. 5x = 3(x+1)). For those that cannot be
solved analytically, use graphical methods to find approximate solutions .
|Explain how the parameters of an exponential or
logarithmic model relate to the data set or situation
being modeled. Find an exponential or logarithmic function to model a given data set or situation.
Solve problems involving exponential growth and decay.
|Solve quadratic-type equations (e.g. e2x - 4 ex+4 =0) by substitution.
|Apply quadratic functions and their graphs in the
context of motion under gravity and simple
|Explain how the parameters of quadratic model
relate to the data set or situation being modeled. Find
a quadratic function to model a given data set or situation.
|Given a polynomial function whose roots are known
or can be calculated, find the intervals on which
the function’s values are positive and those where it is negative .
|Solve polynomial equations and inequalities of
degree greater than or equal to three. Graph
polynomial functions given in factored form using zeros and their multiplicities, testing the sign-on
intervals and analyzing the function’s large-scale behavior.
|Know and apply fundamental facts about
polynomials: the Remainder Theorem, the Factor Theorem,
and the Fundamental Theorem of Algebra.
|Rational Functions and Difference Quotients
|Solve equations and inequalities involving
rational functions. Graph rational functions given in
factored form using zeros, identifying asymptotes, analyzing their behavior for large x values, and
|Given vertical and horizontal asymptotes, find an
expression for a rational function with these
|Know and apply the definition and geometric
interpretation of difference quotient. Simplify
difference quotients and interpret difference quotients as rates of change and slopes of secant lines.
|Define (using the unit circle), graph, and use
all trigonometric function of any angle. Convert between
radian and degree measure. Calculate arc lengths in given circles.
|5.2, 3 (degrees)
6.2 (unit circle)
6.1 (convert arc
|Graph transformations of the sine and cosine
functions (involving changes in amplitude, period,
midline, and phase) and explain the relationship between constants in the formula and transformed
|Know basic properties of the inverse
trigonometric functions sin-1 x, cos-1 x, tan-1 x, including their
domains and ranges. Recognize their graphs.
|Know the basic trigonometric identities for sine,
cosine and tangent (e.g., the Pythagorean identities,
sum and difference formulas, co-functions relationships, double-angle and half-angle formulas).
|5.2, 7.1, 3, 4
|Solve trigonometric equations using basic identities and inverse trigonometric functions.
|Prove trigonometric identities and derive some of
the basic ones (e.g., double-angle formula from sum
and difference formulas, half-angle formula from double-angle formula, etc).
|7.2 - 4
|Find a sinusoidal function to model a given data
set or situation and explain how the parameters of the
model relate to the data set or situation.
|Vectors, Matrices, and Systems of Equations
|Perform operations (addition, subtraction, and
multiplication by scalars) on vectors in the plane. Solve
applied problems using vectors.
|Know and apply the algebraic and geometric definitions of the dot product of vectors.
|Know the definitions of matrix addition and
multiplication. Add, subtract, and multiply matrices.
Multiply a vector by a matrix.
|Represent rotations of the plane as matrices and apply to find the equations of rotated conics.
|Define the inverse of a matrix and compute the
inverse of two-by-two and three-by-three matrices
when they exist.
|Explain the role of determinates in solving
systems of linear equations using matrices and compute
determinants of two-by-two and three-by-three matrices.
|Write systems of two and three linear equations
in matrix form. Solve such systems using Gaussian
elimination or inverse matrices.
|Represent and solve systems of inequalities in
two variables and apply these methods in linear
programming situations to solve problems
|Sequences, Series, and Mathematical Induction
|Know, explain, and use sigma and factorial notation.
|Given an arithmetic, geometric, or recursively
defined sequence, write an expression for the nth term
when possible. Write a particular term of a sequence when given the nth term.
|Understand, explain, and use the formulas for the sums of finite arithmetic and geometric sequences.
|Compute the sums of infinite geometric series.
Understand and apply the convergence criterion for
|Understand and explain the principle of
mathematical induction and prove statements using
|Prove the binomial theorem using mathematical
induction. Show its relationships to Pascal’s triangle
and to combinations. Use the binomial theorem to find terms in the expansion of a binomial to a
power greater than 3.
|Polar coordinates , Parameterizations, and Conic Sections
|Convert between polar and rectangular coordinates. Graph functions given in polar coordinates.
|Write complex numbers in polar form. Know and use De Moivre’s Theorem.
|Evaluate parametric equations for given values of the parameter.
|Convert between parametric and rectangular forms of equations.
|Graph curves described by parametric equations and find parametric equations for a given graph.
|Use parametric equations in applied contexts
(e.g., orbits and projectiles) to model situations and
|Know, explain and apply to locus definitions of
parabolas, ellipses, and hyperbolas and recognize
these conic sections in applied situations.
|10.1 – 3
|Identify parabolas, ellipses, and hyperbolas from
equations, write the equations in standard form, and
sketch an appropriate graph of the conic section.
|Derive the equation for a conic section from
given geometric information (e.g., find the equation of an
ellipse given its two axes). Identify key characteristics (e.g., foci and asymptotes) of a conic section
from its equation or graph.
|10.1 – 4
|Identify conic sections whose equations are in polar or parametric form.
All Chapters and Sections quoted come from Precalculus by Lial, Hornsby and Schneider, 3rd Edtion.